Supporting Structure for Curved Envelope Geometries

ABSTRACT

A support structure for curved envelope geometries, such as buildings and shipbuilding, that at least sectionally approximates a freeform surface, includes longitudinal connection elements and surface elements spanned by the connection elements. The surface elements are implemented as single-curved strip elements whose curvature runs in the longitudinal direction of the strip elements. Adjacent strip elements are connected to one another along their longitudinal edges via the longitudinal connection elements.

CROSS REFERENCE TO RELATED APPLICATIONS

This application is a continuation of International Application No.PCT/EP2009/057805, filed on Jun. 23, 2009, entitled “Support Structurefor Curved Envelope Geometries,” which claims priority under 35 U.S.C.§119(a)-(d) to Application No. AT 1007/2008 filed on Jun. 24, 2008,entitled “Support Structure for Curved Envelope Geometries,” the entirecontents of which are hereby incorporated by reference.

FIELD OF THE INVENTION

The invention relates to a support structure for curved envelopegeometries, in particular in buildings and shipbuilding, the curvedenvelope geometry at least sectionally approximating a freeform surface,comprising connection elements and surface elements spanned by theconnection elements and to methods for fixing such a support structure.

BACKGROUND

Curved envelope geometries of this type are used in construction or inshipbuilding to implement freeform surfaces, in which the curvature isdifferent in two different spatial directions, for example, in buildingswith domes, or also more complex surface shapes. Freeform surfaces ofthis type are also referred to as non-developable surfaces, or asdouble-curved or triple-curved surfaces, and are initially drafted ascontinuous surfaces in the computer model in the course of thearchitectonic drafting. In the construction implementation, thecontinuous freeform surfaces are to be approximated by a plurality ofindividual surface elements, which are held in a support structure.Thus, for example, it is also possible to implement complex freeformsurfaces having multilayered level glass elements, for example, whichare fastened above, between, or below a support structure made of steel,for example.

For the embodiment of the surface elements, there are two fundamentalpossibilities. On the one hand, the attempt can be made to implement theindividual surface elements as planar, i.e., as level surface elements.In this case, the support structure is formed from individual connectionelements, which are each assembled into polygons, for example,triangles, squares, hexagons, etc. The polygons span the supportstructure, the connection elements being implemented as girders whichmeet in node areas, where they are fastened to one another. Aparticularly advantageous embodiment of such a support structure isdescribed in Austrian Patent Number 503,021, in which a torsion-freeimplementation of the individual girders is made possible in particular.This embodiment has the disadvantage, however, that sometimes very manyconnection elements are required, which increase the costs, on the onehand, and also have aesthetic disadvantages, in particular in buildings,on the other hand, since the visual impression of the “lightness” of theenvelope geometry is lost due to the plurality of connection elements.

Another possibility for approximating freeform surfaces by individualsurface elements further comprises approximating the freeform surface bycurved surface elements. In this case, methods of approximating afreeform surface with the aid of double-curved surface elements areknown, which have substantial disadvantages in practice, however, sothat the material selection for the surface elements is subject torestrictions in this case because of the required deformability indouble-curved surfaces, for example. Furthermore, it is typically notpossible to find a distribution of the connection elements with the aidof double-curved surface elements in which the connection elementsimplemented as girders do not have to be subjected to torsion in thegeometrical meaning in the course of the installation between two nodeareas, i.e., a twisting of the longitudinal axis in the node area, forexample. Only connection elements having circular cross-section may bearrayed on one another “torsion-free,” in the geometrical meaning. Ifnoncircular cross-sections are used, up to this point a torsion (in thegeometrical meaning) has arisen in the node area up to this point. Thisresults in aesthetically and statically unsatisfactory node areas.Rather, the problem also results therefrom that multilayer structurescannot be implemented or can only be implemented with substantialadditional outlay. Therefore, a separate support system must be providedfor each layer, which in turn multiplies the material costs and theinstallation effort.

SUMMARY

It is therefore the object of the invention to find a structuralimplementation of freeform surfaces which reduces the technical andeconomic requirements and satisfies aesthetic demands. In particular,installation effort and costs are to be kept as low as possible. Afurther object of the invention is that the support structure for theapproximation of freeform surfaces also offers the possibility of aproblem-free multilayer structure, i.e., the installation of multiplesurface elements offset in parallel. Furthermore, the number ofconnection elements is to be reduced in comparison to known supportstructures based on planar surface elements in triangular, square, orhexagonal form.

Described herein is a support structure for curved envelope geometries,in particular in buildings and shipbuilding, the curved envelopegeometry at least sectionally approximating a freeform surface,comprising connection elements and surface elements spanned by theconnection elements. According to the invention, surface elements areprovided in this case which are implemented as single-curved stripelements, whose curvature runs in each case in the longitudinaldirection of the strip elements, each two strip elements being connectedto one another along their longitudinal edges via longitudinalconnection elements. A buckle results in the area of the longitudinalconnection elements between each two adjoining strip elements, i.e., inthe mathematical meaning, a discrete transition which is kept as smallas possible during the design of the strip elements, in order to havethe best possible approximation of the envelope geometry of the freeformsurface, and thus provide the impression of a continuous envelopegeometry. An uninterrupted array of strip elements is thus implemented,which approximate the freeform surface in their entirety. The way inwhich suitable strip elements are ascertained will be described ingreater detail hereafter. The curvature of the strip elements can bedescribed by lines of curvature, which therefore run in the longitudinaldirection of the strip elements. Because of the single-curved embodimentof the strip elements, generatrixes running transversal to the lines ofcurvature further exist, which are linear. Transverse connectionelements may optionally be additionally provided transversely to thestrip elements, for example, along generatrixes of the strip elements,the strip element also being able to be implemented as interrupted inthe area of the transverse connection elements, so that a strip elementis divided into individual panels.

Such an approximation of freeform surfaces with the aid of single-curvedsurface elements which are uninterruptedly arrayed on one another isalso referred to hereafter as the “strip model.” In the context of themathematical modeling of such strip models, the curves along which thestrip elements are arrayed on one another are also referred to as “edgecurves.” These edge curves correspond to the longitudinal edges of thestrip elements in their structural implementation, along which thelongitudinal connection elements are situated according to theinvention.

In the scope of the invention, it has surprisingly been established thatsuch strip models may also be found for complicated freeform surfaces,and allow an implementation of a support structure according to theinvention. In the scope of the present invention, the term “freeformsurface” is therefore to be understood in particular as a surface whichmeets the following conditions. It is

-   -   a double-curved surface, and    -   it has no kinematic generation by movement (possibly including        scaling) of a curve in such a way that discrete layers of the        moved curve may be used as edge curves of a strip model.

As an example of surfaces which are therefore not freeform surfacesaccording to the invention, surfaces of revolution may be mentioned. Afurther example are the sliding surfaces, generated by displacement of acurve along another curve.

According to an advantageous embodiment of the invention, it can furtherbe provided that the strip elements are implemented so that a family ofgeneratrixes exist in the transverse direction of the strip element, thegeneratrixes enclosing the same angle in each case with the twolongitudinal edges of the strip element. The design of the stripelements is performed via mathematical optimization methods, which arealso referred to hereafter as the “circular model,” because a circle,which touches the two longitudinal edges in the endpoints of thegeneratrix, results upon fulfillment of this condition in the tangentialplane of the generatrix. This will be described in greater detailhereafter.

Alternatively or additionally, it can also be provided that the stripelements are implemented so that a family of generatrixes exist in thetransverse direction of the strip element, in each case a generatrix oftwo adjacent strip elements, which actually intersect or intersect intheir imaginary extension, enclosing the same angle with the tangent atthe actual or imaginary shared longitudinal edge in their point ofintersection. The design of the strip elements is performed viamathematical optimization methods, which are also referred to hereafteras the “conical model,” because a right cone results upon fulfillment ofthis condition in the tangential plane of the generatrix, whose tip liesin the point of intersection of the two generatrixes, and the twoadjacent strip elements touch along the intersecting generatrixes. Thiswill also be discussed in greater detail hereafter. The simultaneousfulfillment of the two mentioned conditions is also conceivable, in thatthey may be required as “soft” secondary conditions, mathematicaloptimization methods of this type also being referred to hereafter as“approximative curved strip models.”

According to an advantageous refinement of the invention, at least twoenvelope geometries which are spaced apart from one another may beprovided, a strip element of a second envelope geometry being formed byparallel displacement of a strip element of a first envelope geometry.Specifically, if one of the two or also both of the above-mentionedconditions are fulfilled, an uninterrupted network of strip elementsaccording to the invention can be displaced in parallel to form afurther uninterrupted network of strip elements. This paralleldisplaceability is also referred to hereafter as “offset.” Correspondinggeneratrixes of two parallel displaced strip elements, and correspondinglongitudinal edge tangents, are parallel.

With respect to the embodiment of the longitudinal connection elements,it can be provided that the longitudinal connection elements areimplemented as cuboid, their height corresponding to the distance of twolongitudinal edges lying one above another. Specifically, correspondinglongitudinal edges of two strip elements according to the inventionwhich are displaced in parallel may be connected by single-curvedsurfaces, on the basis of which corresponding longitudinal connectionelements may be easily manufactured. Transverse connection elements mayalso be implemented as cuboid between generatrixes, which are assignedto one another, of two strip elements displaced in parallel, the heightof the connection elements corresponding to the distance of therespective generatrixes. These transverse connection elements are notcurved, and the occurring nodes between longitudinal girder elements andtransverse connection elements are torsion-free. This allows anembodiment of the support structure according to the invention which issimpler overall.

Furthermore, the invention relates to a method for establishing asupport structure for curved envelope geometries, in particular inbuildings and shipbuilding, comprising connection elements and surfaceelements spanned by the connection elements, the curved envelopegeometry being at least sectionally approximated by a freeform surface.In this case, it is provided according to the invention that the surfaceelements are implemented as single-curved strip elements adjoining oneanother along their respective longitudinal edges, and the longitudinalconnection elements running along the shared longitudinal edges of twostrip elements adjoining one another are implemented so that they followthe course of the respective shared longitudinal edge. The establishmentof the strip elements can concretely be performed, for example, in thatsequences of lines of curvature of the freeform surface are selected,which may each be connected by single-curved strip elements, whosecurvature runs in the longitudinal direction of the strip elements ineach case, and, for the best possible approximation to the freeformsurface, the angles between the generatrixes, which intersect oneanother in their shared longitudinal edge, of a family of generatrixesin the transverse direction of two strip elements adjoining one anotherare minimized. This will be described in greater detail hereafter.

According to an advantageous refinement of the method according to theinvention, it can be provided that the strip elements are implemented sothat a family of generatrixes exists in the transverse direction of thestrip element, the generatrixes each enclosing the same angle with thetwo longitudinal edges of the strip element.

Alternatively or additionally thereto, however, it can also be providedthat the strip elements are implemented so that a family of generatrixesexists in the transverse direction of the strip element, one generatrixof two adjacent strip elements, which actually intersect or intersect intheir imaginary extension, enclosing the same angle in each case withthe tangent on the actual or imaginary shared longitudinal edge in theirpoint of intersection.

If one of these two, or also both conditions are fulfilled, theinvention according to the method can also be refined so that at leasttwo envelope geometries which are spaced apart from one another areestablished, a strip element of a second envelope geometry being formedby parallel displacement of a strip element of a first envelopegeometry.

Finally, it can also be provided in the scope of the method according tothe invention that the longitudinal connection elements are implementedas cuboid, their height corresponding to the distance of twolongitudinal edges lying one above another.

BRIEF DESCRIPTION OF THE DRAWINGS

The invention is explained in greater detail hereafter on the basis ofembodiments with the aid of the appended drawings. In the figures:

FIG. 1 shows a detail of a strip model to explain fundamental concepts,

FIG. 2 a shows the angle conditions and normals along the edge curves ina circular model,

FIG. 2 b shows the angle conditions and normals along the edge curves ina conical model,

FIG. 3 shows a detail of a strip model having offset,

FIGS. 4 a-4 c show figures to explain a geodetic model,

FIGS. 5 a-5 c show figures to explain a cylindrical model for variousgeneratrix directions, the generatrixes only being visualized on everysecond strip for better visibility,

FIG. 6 shows a layer structure made of partially curved, cuboidelements,

FIG. 7 shows a multilayer structure, which implements the main supportstructure via a square network having level planar elements, and placesstrip models thereon, optionally on both sides,

FIG. 8 shows a connection of two strip models, the discrete directionand the continuous direction being selected differently in the twolayers,

FIG. 9 shows a perspective view of a detail of a support structurehaving connection elements, which are implemented as I-beams,

FIG. 10 shows a detail view of the implementation of I-beams in thescope of a support structure according to the invention,

FIG. 11 shows a view of a girder element along an edge curve of ageodetic model,

FIG. 12 shows a girder element having a core part made of multiplerails,

FIG. 13 shows a view of a strip model for the use of level panels inweakly curved areas,

FIG. 14 shows a view of a strip model for the use of double-curvedpanels in strongly curved areas, and

FIG. 15 shows an exemplary view of a freeform surface according to theinvention.

DETAILED DESCRIPTION

The implementation of a freeform surface, as is shown in FIG. 15, forexample, in a support structure according to the invention of abuilding, for example, requires two fundamental steps. Firstly, thedouble-curved freeform surface specified by the planner is to beconverted according to the invention into single-curved strip elementsS, which each adjoin one another continuously along a shared edge curveK. These single-curved strip elements S are also referred to as“developable,” because they may be imaged without distortion on a plane.In a second step, the result of this conversion, which is also referredto hereafter as the “strip model,” or also “curved strip model” of thefreeform surface, is finally to be converted into a support structurewhich can be structurally implemented, the calculated strip elements Soccurring as surface elements of the support structure, and the edgecurves K as longitudinal edges L of these strip elements S. Finally, theconnection elements 1, 2 are to be added, longitudinal connectionelements 1 running along the longitudinal edges L of the strip elementsS, and optionally transverse connection elements 2 running transverselyto the longitudinal connection elements 1. The two steps will beexplained in greater detail hereafter, the developable strip modelestablished according to the invention first being discussed.

Developable strip models may be understood as semi-discrete surfacerepresentations in the following meaning: a family of parameter lines,the edge curves K, occurs discretely, i.e., in a finite number; thesecurves are smooth. The polygons formed by the generatrixes E of thestrip elements S may be understood as the second family of parametercurves, as is obvious on the basis of FIG. 1. This second family isdense, but the curves are “discrete,” i.e., polygons.

From a theoretical viewpoint, these semi-discrete representations lieprecisely between the discrete representations and the smooth surfaces.Concretely, a square network having level meshes, as is described inAustrian Patent Number 503,021, is a discrete version of a strip model.On the continuous side, a so-called conjugated curve network results ona smooth surface. This state of affairs can be used for the actualcalculation in that, depending on the stated object, the optimizationcan be initialized via the discrete version (square network having levelmeshes) or the smoothed version (conjugated network). It is also stilldescribed hereafter that a square network having level meshes and astrip model may be embedded in a single architectonic structure.

In the calculation of an approximation of a given freeform surface by astrip model, it is to be ensured in particular that the edge curves Kavoid the asymptotic directions of the freeform surface, because thesedirections are self-conjugated. Transversal generatrix polygons are thusobtained, and therefore practically usable strip models.

Methods and algorithms for the calculation of strip models are describedin greater detail hereafter. Firstly, several practically importantclasses of strip models will be discussed.

Curved strip models are the semi-discrete counterpart to the networkmade of lines of curvature k on a smooth surface, or also to the knowndiscrete versions of these networks, for example, the circular orconical square networks, as are described in Austrian Patent Number503,021. In a curved strip model, the generatrix polygons areapproximately perpendicular to the edge curves K. If one has a smoothunderlying freeform surface, it can be constructed in that a sequence oflines of curvature k on the freeform surface are selected, and these arethen connected by developable strip elements S. One can also start froma discrete variant and make this into a curved strip model by refiningand optimization. The following discrete identifications of curved stripmodels are suitable for their calculation, reference also being made toFIGS. 2 a and 2 b for this purpose.

Circular model. Each generatrix E of a strip element S meets the twoedge curves K at an equal angle here. Therefore, a circle lying in thetangential plane of the generatrix E results, which touches the two edgecurves K in the endpoints of the generatrix E.

Conical model. One has the same angle in every point p of an edge curveK between the tangents here on the edge curve K and the two generatrixesE of strips ending there. Therefore, a cone having tip p results, whichtouches the adjoining strip elements S along generatrixes E.

Approximative curved strip model. For many practical applications, it issufficient to incorporate the conditions (a) and (b), or a combinationthereof, as “soft” secondary conditions using penalty terms in anoptimization algorithm. The obtained strip models have similarproperties as the models in (a) or (b).

One advantage of curved strip models is the existence of offsets. Togenerate an offset of a model M, a parallel spherical model M_(S) canfirst be calculated. This is a strip model which approximates a sphere S(of radius 1) and has a unique correspondence to M, so thatcorresponding generatrixes E and edge curve tangents of M and M_(S) areparallel. In a circular model M, the edge curves K of M_(S) are on thesphere S. For a conical M, the strips of the parallel model M_(S) of thesphere S are redrawn touching. In the case of an approximative curvedstrip model, M_(S) approximates the sphere S well.

The offsets may now be easily described analytically. In this case, S iscentered in the origin of the employed coordinate system. If p refers tothe coordinate vector of a point of an edge curve K of M and p_(S) tothe associated point of M_(S), then p_(d)=p+d p_(S) is the correspondingpoint of the edge curve K of the offset model M_(d) of M at the distanced. The vector p_(S) is used as the normal vector of M in p. In a conicalmodel, this vector is in the axis of the above-described touching rightcone.

The constant offset distance d is measured as follows: in a circularmodel, it occurs between corresponding points p and p_(d) on edge curvesK of the starting model and the offset. In a conical model M,corresponding generatrixes E and tangential planes of the strip elementsS of M and M_(d) are at the constant distance d (see also FIG. 3).

The ruled surface strips formed by the connection sections ofcorresponding points p and p_(d) between corresponding edge curves K(“normal surfaces”) are developable. The construction advantagesresulting therefrom are described in greater detail hereafter. This isalso true for approximative curved strip models.

Furthermore, it is to be noted that offsets and developable “normalsurfaces” (connection surfaces of edge curves K on base and offset) canalso be calculated for developable arbitrary strip models byoptimization. However, stronger deviations will possibly result in thedistances occurring between base and offset, or the normal surfaces willno longer be approximately perpendicular to base and offset.

The geodetic model, as is shown in FIG. 4 a, represents a special caseof a strip model. A geodetic model is a strip model whose developablestrips follow the geodetic lines of an underlying smooth surface. As aconsequence, the development of the strip elements S is nearly linear.Such models may also be directly identified as follows: in every point pof an edge curve K, the osculating plane of the edge curve K forms equalangles with the tangential planes of the adjoining strip elements S (seealso FIG. 4 b in this regard). Therefore, the edge curve K has the sameabsolute value of the geodetic curvature for the two strip elements Sand the edge curve K is imaged on congruent curves in oppositedirections in the development of the two strip elements S (see FIG. 4c). Geodetic models are well suitable for the coverage of curvedfreeform surfaces having long panels, for example, made of wood, whichhave an almost linear development. It can be necessary to cover afreeform surface using multiple geodetic models of varying direction.

Cylindrical models have strip elements S made of general cylindricalsurfaces (see also FIG. 5 a). These models can be constructed by mergingtouching redrawn cylinders of a given freeform surface, as shown inFIGS. 5 a-5 c. The optimization algorithms described hereafter are alsousable for this purpose.

Since cylinders, in particular right cylinders, are producible moreeasily in certain materials (such as glass) than general developablepanels, one will sometimes attempt to employ panels which are ascylindrical as possible. It is not necessary for this purpose to use acylindrical model and to provide a single cylindrical surface per stripelement S. Rather, a general developable strip element S is decomposedinto panels and each panel is approximated by a cylinder (in particulara right cylinder). This is well possible in particular if the recessionpoints of the strip element S are far away from the strip element S. Theapproximation using right cylinders is a standard task of geometricaldata processing and can be performed using known algorithms for surfaceapproximation and registration. The latter method is to be used if onlya finite fixedly predefined number of possible cylinder radii is to bemaintained. This method can be performed perfectly as long as theapproximation lies within the manufacturing tolerances (or thetolerances for cold bending of glass).

Conical models have strip elements S made of general conical surfaces.Similar conditions apply here as in the cylindrical models. Conicalmodels can be constructed by merging touching redrawn cones of a givenfreeform surface.

If one wishes to use right cones as panels for manufacturing reasons,general developable strip elements S can in turn be decomposed intopanels and an approximating right cone can be calculated per panel usingknown methods. It is known from the theory of the developable surfaces(for example, because of the existence of the so-called curved cone),that a segmentation into right cones is well possible.

Finally, models having level edge curves K will be discussed briefly.Upon the implementation of the edge curves K of a strip model inconnection elements 1, 2, specific advantages result during themanufacturing of the connection elements 1, 2 if they lie in planes. Theplanarity of edge curves K can be incorporated into the below-mentionedoptimization. However, the possibility also results of connecting asequence of level sections of a given freeform surface by developablestrip elements S. It is to be noted that a strip model represents asemi-discrete version of a conjugated curve network. Therefore, levelsections are to be avoided, which touch osculating directions of thefreeform surface (these generate turning points in the level sections).

Partially level edge curves K are still simpler to achieve. A verysimple and practical solution is to approximate the edge curves K of anarbitrary strip model by so-called arc splines and then optionallyincorporate new developable strip elements S which are adapted to themodified edge curves.

Possible optimization algorithms for calculating strip models aredescribed hereafter. The calculation of a strip model is performed withthe aid of a numeric optimization algorithm. The ith edge curve K isestimated as a third-degree B spline curve, for example, and havinguniform nodes, as

p _(i)(u)=Σ_(j) B ³(u−j)b _(i,j)

Sequential edge curves K are connected by linear interpolation of thetwo curve representations to form a strip element S:

x _(i)(u,v)=(1−v)p _(i)(u)+vp _(i+1)(u)| x _(i)(u,v)=(1−v)p _(i)(u)+vp_(i+1)(u)| x _(i)(u,v)=(1−v)p _(i)(u)+vp _(i+1)(u)| x _(i)(u,v)=(1−v)p_(i)(u)+vp _(i+1)(u)|

The system thus described of strip elements S made of ruled surfaces issubjected to an optimization, which describes the developability, theproximity to a reference surface, and the smoothness of the resultingmodel. A target function of the following form thus results for theoptimization

F=w ₁ f _(Abw) +w ₂ f _(Flaeche) +w ₃ f _(Rand) +w ₄ f _(glatt)|

The individual terms are defined as follows:

The developability of the ith strip is achieved by minimization of

f _(Abw)=Σ_(i)∫δ_(p) _(i) _(,p) _(i+1) (u)² du|

The integrand identifies the squared distance of the diagonals in thepolygon

p _(i) ,p _(i)+λ_(i) {dot over (p)} _(i) ,p _(i+1)+μ_(i) {dot over (p)}_(i+1) ,p _(i+1)|

Dots indicate derivatives according to the curve parameter u therein.These polygons must result as level in the case of a developablesurface. In order that the deviation receives a practical significancefor the design, the points on the tangents are selected equal to thelength of the generatrix distance, i.e., the following condition is set:

λ_(i) =∥p _(i+1) −p _(i) ∥/∥{dot over (p)} _(i)∥, μ_(i) =∥p _(i+1) −p_(i) ∥/∥{dot over (p)} _(i+1)∥

The proximity to a reference surface R is controlled by

f _(Flaeche)=Σ_(k)dist(x _(k) ,T _(k))²

The points x_(k) are selected sufficiently densely on the strip modeltherein and T_(k) refers to the tangential plane of the referencesurface R in the point of the reference surface R closest to x_(k).Squared distances to tangential planes of the reference surface R arethus minimized. This does not have to be performed precisely in thismanner, but converges more rapidly than squared distances to closestpoints of the reference surface R. To maintain boundaries, tangentst_(k) are used on the given boundaries of the reference surface Rinstead of tangential planes,

f _(Rand)=Σ_(k)dist(x _(k) ,t _(k))²

Finally, it is important above all in an application as in architecture,where aesthetics play a large role, to also achieve the best possiblesmoothness (aesthetics) of the strip model. The term responsible forthis in the optimization is a combination of a term which is responsiblefor the smoothness of the edge curves K,

f _(glatt/1)=Σ_(i) ∫∥{umlaut over (p)} _(i)(u)∥² du

and a term which positively influences the aesthetics of the generatrixpolygons transversal thereto,

f _(glatt/2)=∫(Σ_(i) ∥p _(i+1)−2p _(i) +p _(i−1)∥²)du

Linearized bending energies, i.e., integrals over the squares of thesecond derivatives or their numeric approximations, were used here.However, third derivatives may also be used, or, at best, combinationsof second and third derivatives.

For special models, such as circular or conical models, a term must alsobe added, of course, which contains the characterizing property of therespective model. This is noted as an example for the circular model,

$f_{zirk} = {\sum\limits_{i}{\int{{\langle{{p_{i + 1} - p_{i}},{\frac{{\overset{.}{p}}_{i}}{{\overset{.}{p}}_{i}} + \frac{{\overset{.}{p}}_{i + 1}}{{\overset{.}{p}}_{i + 1}}}}\rangle}^{2}{u}}}}$

A very similar procedure is used in the other cases.

Various known methods of nonlinear optimization may be used for theactual numeric optimization. This is a function of the desired precisionat which the individual requirements are to be achieved. In many cases,it is sufficient to follow the penalty approach described here. Asuitable control of the weights w_(i) in the course of the iterativemethod is to be ensured. However, methods of restricted optimization mayalso be used.

A suitable initialization of the optimization is very important. Thiscan be performed from the discrete side or from the continuous side.Depending on the model type, in the first-mentioned case, the startingversion of the strip model is established using a discrete versionthereof (square network having level meshes, conical network, etc., asdescribed in Austrian Patent Number 503,021). In the second-mentionedcase, a continuous version is used (conjugated network, network of thelines of curvature k, etc.).

With respect to the implementation of a strip model as an architectonicsupport structure, for example, or in boat building or shipbuilding,there are various possibilities for using strip models. These are afunction of the employed materials, costs, existing constructiontechnologies, and also aesthetic and structural considerations. Somepossibilities are listed hereafter, mixed forms are conceivable and arenot explicitly described.

To implement a first variant of a multilayered structure, for example, astrip model M (at best a curved strip model) is used as a startingpoint, as well as an offset M_(d) of this model. Corresponding edgecurves K of base model and offset may be connected by developablesurfaces. A sequence of generatrix polygons may also be selected on thestrip model M, and connected to the corresponding generatrix polygons onM_(d) by plane parts. A layered structure made of partially curved“cuboid” elements thus results, only planes and developable surfacesoccurring as the lateral surfaces thereof. The occurring nodes aretorsion-free. Such a multilayered structure is shown in FIG. 6. AlthoughFIG. 6 only shows two layers, more layers may also be implemented inthis way, of course.

In modern constructions, however, various functions (aesthetics,structure, water tightness, insulation, ventilation, etc.) are oftenseparated in various technical elements. It is not necessary to curveall of these elements. Therefore, a second variant of a multilayeredstructure is proposed, in which the main support structure isimplemented via a square network having level meshes, and strip modelsare placed thereon, optionally on both sides. The natural closerelationship between strip models and square networks having levelmeshes, which is founded in the geometrical state of affairs, makes itobvious to connect these two geometries in a single structure, as shownin FIG. 7. The calculation can be performed starting from a squarenetwork P having level meshes, as was described, for example, inAustrian Patent Number 503,021, and an offset P_(d) of P. A strip modelM is constructed from P_(d) by the above-described optimizationapproach, the nodes of P_(d) being interpolated (optionally within agiven tolerance). The optimization can also be used so that constantdistances result between node points of P and the corresponding contactpoints of the strip model M. The main support structure can now beimplemented by the torsion-free design of linear connection elements 1,2, which are connected to P.

In a third variant of a multilayered structure, in the abovemultilayered structure, the network P can also be left out in the finalembodiment, the strip models being designed in various directions onboth sides of P (see FIG. 8). It is of interest in this case that thelinear elements are situated externally and internally in variousdirections, and a structurally noteworthy, completely novel constructionthus results.

The construction of connection elements 1, 2 and panel fixations, whichconsider the edge curves K and developable surfaces connected thereto,will be discussed in greater detail hereafter. The edge curves K of thestrip models occur in this case as longitudinal edges L of the supportstructure. For the construction of the support structure for a freeformsurface based on a strip model, for example, in an architectonicapplication, the geometric properties of the strip model can be used forthe design and manufacturing of the connection elements 1, 2, forexample, as girders. This is essentially based on developable surfaces,which are linked to the edge curves of a strip model.

One possibility for implementing the connection elements 1, 2 is in theform of I-beams, for example. An I-beam comprises three parts, namely anupper flange 3 and a lower flange 4, which absorb axial pressure aboveall, and a core 5, which bears shear load above all (see FIGS. 9 and10). It is possible to manufacture a curved girder separately from thesethree parts. It is advantageous if all three parts are developable.First the development of the connection element 1, 2 is cut out of thedesired material (e.g., steel), and it is then bent into the finalshape. The moment of the three connected parts is higher than that ofeach individual part, and the separate manufacturing is thereforeadvantageous.

As is obvious from FIG. 10, the upper flange 3 and the lower flange 4lie approximately tangentially on offsets of the fundamental stripmodel, and the core 5 lies normal thereto. The strip elements S have acoating 7, which is to be adapted to the required structural demands. Itis possible that in the structural implementation of a support structureaccording to the invention, the strip elements S do not entirely abutone another, but rather two adjoining strip elements S are held in areceptacle 6 at a small distance from one another. The receptacle 6 alsoensures the required water tightness, for example. In FIG. 10, theimaginary extensions S′ of two adjoining strip elements S are also shownby dashed lines, which intersect in the longitudinal edge L. Thislongitudinal edge L corresponds to the edge curve K of the fundamentalstrip model.

Curved strip models have an array of advantages for this purpose.Offsets exist, and a developable surface is obtained by connectingcorresponding edge curves K of base and offset. This can be used as thecore 5 of the connection elements 1, 2. The two flange parts 3, 4 areimplemented approximately tangentially on suitable offsets of the stripmodel. It is advantageous for the bending of the flange parts 3, 4 thatthe generatrixes E of the fundamental strip model are approximatelyperpendicular to the edge.

If a connection element 1, 2 follows an edge curve K of a geodetic stripmodel, the two flange parts 3, 4 have a linear development. However, thecore 5 can then only be implemented as developable if it does not runperpendicular to the flanges 3, 4. A connection of the two flange parts3, 4 by a system of linear connection bars in a crossed configurationcan then replace the core 5, as shown in FIG. 11. For a model havinglevel edge curves K, the core 5 can also be selected in this plane inany case, when the angle of intersection between the fundamentalfreeform surface and the plane is not too flat at any point. This anglealso occurs as the angle between the flange parts 3, 4 and the core 5.

Finally, the core 5 of a connection element 1, 2 can be manufacturedfrom multiple rail parts 5 a, which slide on one another, and which arefirst fixed in the final position (FIG. 12). Depending on the strain ofthe envelope geometry, a H-beam can also be used better than an I-beam.Completely similar considerations apply here. It is advantageous if thethree parts of a H-beam do not have to be bent strongly, i.e., result asalmost level. In particular in models having level edge curves K, thisvariant can be of interest.

A further support structure which can be linked well with strip modelsis given by tubular girders (having circular cross-section). Thecenterline of the tube is laid out at a constant distance to the model(for example, edge curve K on an offset). It is advantageous here if thecenterline is formed by a smoothly arrayed sequence of circularsegments, because these tubes are easily available standard parts.

Finally, the possibility is also to be discussed of using level panels 8in weakly curved areas (see FIG. 13). Questions of cost play animportant role in the selection of panel types for a given freeformgeometry. It is therefore advisable to provide level panels 8 in thoseareas in which the developable panel deviates only very slightly from aplane. The close theoretical and algorithmic relationship between modelsmade of developable strips and level square networks makes it verysimple to install level panels 8 in the weakly curved areas. A criterionfor the selection of a level surface element instead of a very weaklysingle-curved panel 10 is the buckle angle occurring along thelongitudinal edges L of adjoining panels. If this is less than a barrieras a function of the aesthetic demands and the surface behavior of thematerial (reflection properties), level panels 8 will be provided. Allabove-mentioned multilayered structures and girder designs may also beaccordingly applied in this hybrid case (level and single-curvedpanels).

In strongly curved areas, in contrast, double-curved panels 9 may alsobe used (see FIG. 14). The support structure according to the inventiondoes not preclude double-curved panels 9 also being incorporatedsectionally, if the curvature is so strong in both main directions inindividual areas that the buckle angles occurring along the edge curvesK of a strip model become too large. The geometry can fundamentally betaken directly from the underlying freeform geometry. If themanufacturing of the panels prefers specific types of double-curvedpanels 9, instead of the given freeform geometry, a double-curved panel9 can also be used, which is of this easily producible type, andapproximates the given geometry in the scope of the desired tolerances.Overall, all above-mentioned multilayered structures and girder designsare also to be accordingly applied in the hybrid case (level panels 8,single-curved panels 10, and double-curved panels 9).

It is therefore obvious how a support structure according to theinvention can be applied in manifold ways for the structuralimplementation of freeform surfaces. The implementation of freeformsurfaces according to the invention reduces the technical and economicrequirements, and nonetheless satisfies aesthetic demands. Inparticular, installation effort and costs may be kept as small aspossible. Furthermore, the possibility also exists of a problem-freemultilayered structure, i.e., a parallel offset installation of multiplelevel surface elements. Furthermore, the number of connection elements1, 2 is reduced in comparison to known support structures based onplanar surface elements in triangular, square, or hexagonal form.

1. A support structure for curved envelope geometries, the curvedenvelope geometry at least sectionally approximating a freeform surface,comprising: longitudinal connection elements; and surface elementsspanned by the connection elements, wherein the surface elements areimplemented as single-curved strip elements whose curvature runs in thelongitudinal direction of the strip elements, and wherein pairs of stripelements are connected to one another along their longitudinal edges viathe longitudinal connection elements.
 2. The support structure accordingto claim 1, wherein the strip elements are implemented such that afamily of generatrixes exists in a transverse direction of the stripelements, the generatrixes each enclosing the same angle with the twolongitudinal edges of the strip element.
 3. The support structureaccording to claim 2, wherein at least two envelope geometries which arespaced apart from one another are provided, a strip element of a secondenvelope geometry being formed by parallel displacement of a stripelement of a first envelope geometry.
 4. The support structure accordingto claim 3, wherein the longitudinal connection elements are implementedas cuboid, and their transverse extension perpendicular to theirlongitudinal axis corresponds to a distance of two longitudinal edgeslying one above another.
 5. The support structure according to claim 1,wherein the strip elements are implemented such that a family ofgeneratrixes exists in a transverse direction of the strip element,wherein one generatrix of two adjacent strip elements that actuallyintersect or intersect in their imaginary extension, encloses the sameangle with tangents on an actual or imaginary shared longitudinal edgein their point of intersection.
 6. The support structure according toclaim 5, wherein at least two envelope geometries which are spaced apartfrom one another are provided, a strip element of a second envelopegeometry being formed by parallel displacement of a strip element of afirst envelope geometry.
 7. The support structure according to claim 5,wherein the longitudinal connection elements are implemented as cuboid,and their transverse extension perpendicular to their longitudinal axiscorresponds to a distance of two longitudinal edges lying one aboveanother.
 8. A method for establishing a support structure for curvedenvelope geometries comprising longitudinal connection elements andsurface elements spanned by the connection elements, the curved envelopegeometry at least sectionally approximating a freeform surface, themethod comprising: implementing the surface elements as single-curvedstrip elements that adjoin one another along their respectivelongitudinal edges; and implementing the longitudinal connectionelements running along shared longitudinal edges of two adjoining stripelements so that the longitudinal connection elements follow a course ofthe respective shared longitudinal edge.
 9. The method according toclaim 8, wherein the strip elements are implemented such that a familyof generatrixes exists in the transverse direction of the strip element,the generatrixes enclosing the same angle with two longitudinal edges ofthe strip element.
 10. The method according to claim 9, wherein at leasttwo envelope geometries, which are spaced apart from one another, areestablished, a strip element of a second envelope geometry being formedby parallel displacement of a strip element of a first envelopegeometry.
 11. The method according to claim 9, wherein the longitudinalelements are implemented as cuboid, their height corresponding to adistance of two longitudinal edges lying one above another.
 12. Themethod according to claim 8, wherein the strip elements are implementedsuch that a family of generatrixes exists in a transverse direction ofthe strip elements, wherein one generatrix of two adjacent stripelements that actually intersect or intersect in their imaginaryextension, encloses the same angle with tangents on an actual orimaginary shared longitudinal edge in their point of intersection. 13.The method according to claim 12, wherein at least two envelopegeometries, which are spaced apart from one another, are established, astrip element of a second envelope geometry being formed by paralleldisplacement of a strip element of a first envelope geometry.
 14. Themethod according to claim 12, wherein the longitudinal elements areimplemented as cuboid, their height corresponding to a distance of twolongitudinal edges lying one above another.